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Abstract

In this chapter, one investigates the chaos synchronization of a class of uncertain optical chaotic systems. More precisely, one also presents a systematic approach for designing a fractional-order (FO) sliding mode controller to achieve a rapid, robust, and perfect chaos synchronization. By this robust controller, it is rigorously proven that the associated synchronization error is Mittag-Leffler (or asymptotically) stable. In a numerical simulation framework, this synchronization scheme is tested on many chaotic optical systems taken from the open literature. The obtained results clearly show that the proposed chaos synchronization controller is not only strongly robust with respect to the unavoidable system's uncertainties (as unmodeled dynamics, and parameters' variation and uncertainty) and eventual dynamical external disturbances, but also can significantly reduce the chattering effect.

Introduction

There are many practical reasons for controlling and synchronizing of chaotic systems (Chen & Dong, 1998; Li et al.,2016; Ahmad et al., 2016; Othman et al., 2016; Pham et al., 2016; Li et al., 2014 ; Li et al., 2013). This is why people get more and more attention in the research of the chaotic system theory and its applications. Chaotic systems are nonlinear and deterministic yet long-term unpredictable. This impossibility of predicting the future state of a deterministic system is due to the sensitivity of chaotic orbits (Li & Chen, 2004). A chaotic behavior is characterized by the exponential divergence of initially nearby trajectories, or equivalendy, by sensitive dependence on initial conditions. Exponential divergence is quantified by Lyapunov exponents. Note that the chaotic comportment can appear in both dissipative systems and conservative systems. In recent literature, it also was revealed that many fractional-order (FO) systems can chaotically behave, for instance: FO Rössler system (Li & Chen, 2004), FO Arneodo system (Lu, 2005), FO Lü system (Deng & Li, 2005), FO Lorenz system (Grigorenko & Grigorenko, 2003).

After the finding of master-slave synchronization by Pecora and Carroll in 1990, the study of chaotic synchronization became popular (Hasler et al., 1998). The potential applications of chaos synchronization in secure communication systems, which were experimentally established in 1991, engaged the interest of the physician and engineer communities, and it has become an active topic of research since. Many different aspects of chaos synchronization were studied: synchronization in unidirectionally and bidirectionally coupled systems (Hale, 1997), (Ushio, 1995), anti-phase synchronization (Ushio, 1995; Rosenblum et al., 1996), partial synchronization (Hasler et al., 1998) and generalized synchronization (Abarbanel et al., 1996; Kocarev & Parlitz, 1996), and so on. This phenomenon has been observed in several physical systems, e.g. electrical and mechanical systems (Chua & Itoh, 1993), optical and laser systems (Terry & Thornburg Jr, 2009; Behnia et al., 2013), biological systems (Vaidyanathan, 2015) and Josephson junctions (Shahverdiev et al., 2014). Chaos synchronization has also found a number of applications in control theory (Pyragas, 1992), parameter estimation from time series (Maybhate & Amritkar, 1999), secret communication, information sciences, optimization problems, and in some connected nonlinear areas (Zhang et al.,2016a). A good review of developments in the theoretical and experimental study of synchronized chaotic systems can be found in (Aziz-Alaoui, 2005).